Area from random length
Let there be a line segment of exactly 7 cm. I split it at a random point along its length to form two smaller segments, which I then use as the legs of a right triangle.
What is the probability that the triangle thus formed has an area of 6 or more square cm?
Express your answer as a decimal to three decimal places.
The answer is 0.143.
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1 solution
Let the two segments formed have a length of x and 7 − x cm respectively. If they are used as legs of a right triangle, the area of the triangle is 1 / 2 ∗ x ∗ ( 7 − x ) square cm.
So x ∗ ( 7 − x ) has to be 12 or more. This restricts x to the interval [3, 4], which is 1/7 of the length of the segment. So the probability is 1/7 = 0.142857.